Q.46 - Calculus

Question

Graph the cardioids $r = 1 + \cos \theta , r = 2(1 + \cos \theta ), and r = 3(1 + \cos \theta )$ in the same polar coordinate system. What are the relationships among these three graphs? How does changing the constant $k$ in the graph of the cardioid$r = k(1 + \cos \theta )$ affect the graph? What happens when $k<0$? 

Step-by-Step Solution

Answer

The graph is :

Step 1. Given Information.

The given curves are :

$r = 1 + \cos \theta , r = 2(1 + \cos \theta ), and r = 3(1 + \cos \theta )$.

Step 2. Graphically.

The graph of the curves will be :

Step 3. Theoretically.

The given equations are :

$r=k\left(1+\cos \theta \right)$

In equations, the sign$\left(+ and -\right)$ affects the direction of the cardioid. Changing the value of k changes the size of the cardioid but does not affect its shape. The larger the value of k, the larger the resulting cardioid. If the value of $k<0$ , then smaller will be the cardioid.